Method and system for simulating implied volatility surface for basket option pricing

ABSTRACT

A method and system for simulating changes in volatility for a price of a particular option on an underlying financial instrument is disclosed. A volatility surface model having at least one surface parameter is provided along with a set of volatilities for a plurality of options on the underlying financial instrument. The set of volatilities is analyzed to determine an initial value for each surface parameter which, when used in the surface model, defines a surface approximating the set of volatilities. The values of the surface parameters are then evolved using an appropriate evolution function. A volatility value for a particular option is extracted from the volatility surface defined by the evolved surface parameter values. The extracted volatility value can then be used in an option pricing model to provide a price of the particular option.

CROSS-REFERENCE TO RELATED APPLICATIONS

This application is a continuation of and claims priority under 35U.S.C. §120 to U.S. patent application Ser. No. 12/210,147 filed on Sep.12, 2008 and entitled “Method and System for Simulating ImpliedVolatility Surface for Basket Option Pricing,” which in turn claimspriority under 35 U.S.C. §120 to and is a continuation of U.S. patentapplication Ser. No. 10/160,469 filed May 31, 2002 and entitled “Methodand System for Simulating Implied Volatility Surfaces for Basket OptionPricing,” and is also a CIP of U.S. patent application Ser. No.09/896,488 filed Jun. 29, 2001 and entitled “Method and System forSimulating Implied Volatility Surfaces for Use in Option PricingSimulations.” The entire content of each application is herebyincorporated by reference.

FIELD OF THE INVENTION

This invention is related to a method and system for measuring marketand credit risk and, more particularly, to an improved method for thesimulating the evolution of a volatility surface for basket and othermulti-component options for use in simulating the performance of thebasket option.

BACKGROUND

A significant consideration which must be faced by financialinstitutions (and individual investors) is the potential risk of futurelosses which is inherent in a given financial position, such as aportfolio. There are various ways for measuring potential future riskwhich are used under different circumstances. One commonly acceptedmeasure of risk is the value at risk (“VAR”) of a particular financialportfolio. The VAR of a portfolio indicates the portfolio's market riskat a given percentile. In other words, the VAR is the greatest possibleloss that the institution may expect in the portfolio in question with acertain given degree of probability during a certain future period oftime. For example, a VAR equal to the loss at the 99^(th) percentile ofrisk indicates that there is only a 1% chance that the loss will begreater than the VAR during the time frame of interest.

Generally, financial institutions maintain a certain percentage of theVAR in reserve as a contingency to cover possible losses in theportfolio in a predetermined upcoming time period. It is important thatthe VAR estimate be accurate. If an estimate of the VAR is too low,there is a possibility that insufficient funds will be available tocover losses in a worst-case scenario. Overestimating the VAR is alsoundesirable because funds set aside to cover the VAR are not availablefor other uses.

To determine the VAR for a portfolio, one or more models whichincorporate various risk factors are used to simulate the price of eachinstrument in the portfolio a large number of times using an appropriatemodel. The model characterizes the price of the instrument on the basisof one or more risk factors, which can be broadly considered to be amarket factor which is derived from tradable instruments and which canbe used to predict or simulate the changes in price of a giveninstrument. The risk factors used in a given model are dependent on thetype of financial instrument at issue and the complexity of the model.Typical risk factors include implied volatilities, prices of underlyingstocks, discount rates, loan rates, and foreign exchange rates.Simulation involves varying the value of the risk factors in a model andthen using the model to calculate instrument prices in accordance withthe selected risk factor values. The resulting price distributions areaggregated to produce a value distribution for the portfolio. The VARfor the portfolio is determined by analyzing this distribution.

A particular class of instrument which is simulated is an option. Unlikesimple securities, the price of an option, and other derivativeinstruments, is dependant upon the price of the underlying asset price,the volatility of changes in the underlying asset price, and possiblychanges in various other option parameters, such as the time forexpiration. An option can be characterized according to its strike priceand the date it expires and the volatility of the option price isrelated to both of these factors. Sensitivity of the option volatilityto these effects are commonly referred to skew and term. Measures of thevolatility for a set of options can be combined to produce a volatilitysurface. For example, FIG. 1 is a graph of the implied volatilitysurface for S&P 500 index options as a function of strike level and termto expiration on Sep. 27, 1995.

The volatility surface can be used to extract volatility values for agiven option during simulation. The extracted volatility value isapplied to an option pricing model which provides simulated optionprices. These prices can be analyzed to make predictions about risk,such as the VAR of a portfolio containing options. The volatilitysurface is not static, but changes on a day-to-day basis. Thus, in orderto make risk management decisions and for other purposes, changes in thevolatility surface need to be simulated as well.

Various techniques can be used to simulate the volatility surface overtime. In general financial simulations, two simulation techniques areconventionally used: parametric simulation and historical simulation andvariations of these techniques can be applied to simulate volatilities.

In a parametric simulation, the change in value of a given factor ismodeled according to a stochastic or random function responsive to anoise component ε is a noise component. During simulation, a suitablevolatility surface can be used to extract a starting volatility valuefor the options to be simulated and this value then varied in accordancewith randomly selected values of noise over the course of a simulation.

Although parametric simulation is flexible and permits the modelparameters to be adjusted to be risk neutral, conventional techniquesutilize a normal distribution for the random noise variations and mustexplicitly model probability distribution “fat-tails” which occur inreal life in order to compensate for the lack of this feature in thenormal distribution. In addition, cross-correlations between variousfactors must be expressly represented in a variance-covariance matrix.The correlations between factors can vary depending on the circumstancesand detecting these variations and compensating is difficult and cangreatly complicate the modeling process. Moreover, the computationalcost of determining the cross-correlations grows quadradically with thenumber of factors making it difficult to process models with largenumbers of factors.

An alternative to parametric simulation is historical simulation. In ahistorical simulation, a historical record of data is analyzed todetermine the actual factor values and these values are then selected atrandom during simulation. This approach is extremely simple and canaccurately capture cross-correlations, volatilities, and fat-tail eventdistributions. However, this method is limited because the statisticaldistribution of values is restricted to the specific historical sequencewhich occurred. In addition, historical data may be missing ornon-existent, particularly for newly developed instruments or riskfactors, and the historical simulation is generally not risk neutral.

Accordingly, there is a need to provide an improved method forsimulating a volatility surface to determine volatility values duringoption pricing simulation.

It would be advantageous if such a method captured cross-correlationsand fat-tails without requiring them to be specifically modeled andwhile retaining the advantageous of parametric modeling techniques.

It would also be advantageous if such a method could be extended toother multi-variant factors which are used in option pricing models.

In addition to simulating the performance of options based upon singlesecurities, it is also useful to simulate the performance of basketoptions, options based on various indexes, and other options based onthe performance of multiple underlying securities. Conventional practiceis to use a regression analysis to determine volatilities for basketoptions for use during simulation. However, this is computationally veryexpensive.

It would be therefore be of further advantage to provide an improvedmethod of determining volatilities for basket and other multi-securityoptions for use in simulation and other applications.

SUMMARY OF THE INVENTION

These and other needs are met by the present invention wherein optionvolatility is simulated by defining a parameterized volatility surfaceand then evolving the surface parameters in accordance with historicaldata during the simulation. In particular, a volatility surface model isdefined by a series of surface parameters β. The initial values of thesurface parameters are determined by regressing the set of initialoption volatility data relative to expiration time vs. delta or otherappropriate axes. The model is calibrated to determine the offset of thestarting option volatilities from the value provided by the initialsurface model.

At each “tick” of the simulation, the beta parameter values defining thevolatility surface are adjusted according to a function which provides anext beta value based upon the present beta value and a noise-varyingmeasure of the beta volatility. The beta volatility can be determined byanalyzing a time-series of beta values from volatility surfaces derivedfrom historical data or estimated through other means. The new betaparameter values are then applied to the surface model to define asimulated volatility surface which is used to extract a volatility valuefor an option during simulation. The extracted value is adjusted inaccordance with the calibration data and the calibrated simulatedvolatility value is applied to the pricing model.

Various techniques can be used to simulate the noise-varying volatilityof the beta parameters. Preferably, and according to a further aspect ofthe invention, the noise variations in the beta volatility are selectedfrom a set of risk-neutral bootstrapped residual values generatedthrough analysis of a time-varying sequence of beta values fromvolatility surfaces fit to historical data.

According to a further aspect of the invention, the beta surfaceparameter values derived for individual instruments can then be combinedto determine the surface parameters for a volatility surface model ofthe basket directly from the volatility model surface parameters for thesecurities that comprise the basket. As a result, once the surfaceparameters for the individual securities have been generated, thesurface parameters for basket options based on any set of thosesecurities can be easily and quickly generated. Exchange rate volatilitycan also be accounted for to allow simplified simulation of optionbaskets based upon instruments priced in currencies other than thebasket currency.

BRIEF DESCRIPTION OF THE FIGURES

The foregoing and other features of the present invention will be morereadily apparent from the following detailed description and drawings ofillustrative embodiments of the invention in which:

FIG. 1 is a graph of a sample volatility surface;

FIG. 2 is a graph of a set of volatility points for various optionsplotted against the corresponding T and Δ axis;

FIG. 3 shows an implied volatility surface determined in accordance withthe invention for the set of volatility data points of FIG. 2;

FIG. 4 is a flowchart of a method for simulating a volatility surface inaccordance with the present invention; and

FIG. 5 is a flow diagram of a process for simulating option pricessystem in accordance with the present invention.

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS

The present invention is directed to an improved technique forsimulating the time-evolution of a risk factor value which is dependantupon two or more variables. This invention will be illustrated withreference to simulating the performance of derivative instruments with arisk factor dependant upon multiple factors, and, in particular, thevolatility surface for options. Option prices have a volatility that isdependant upon both the price of the underlying security and the timeremaining before the option expires. The volatility for the variousoptions which derive from a given security can be represented as avolatility surface and the present methods provide an improved techniquefor simulating the evolution of the volatility surface for use in, e.g.,risk analysis simulations. The methodology can be applied to other typesof derivative instruments and more generally to simulation models whichhave risk factors dependant upon multiple factors which can be modeledas “multi-dimensional surfaces”, such as volumes, or higher dimensionalconstructs.

An option can be characterized according to its strike price and thedate it expires and the volatility of the option price is related toboth of these factors. The ratio between the change in option price Pand the security price S is conventionally expressed as “delta”:

$\begin{matrix}{\Delta = \frac{\partial P}{\partial S}} & \left( {{Equ}.\mspace{14mu} 1} \right)\end{matrix}$One method of specifying a volatility surface is with reference to deltavs. the term T remaining for an option, e.g., σ(T,Δ). The use of deltaprovides a dimensionless value which simplifies comparisons betweendifferent options. However, other variables for the surface σ(x,y) canbe alternatively used.

Initially, historical data for options of a given security is analyzedto determine (or otherwise select) an implied volatility σ_(imp) foreach option of interest at a starting point of the simulation, e.g.,beginning from the most recent closing prices. The volatility pointsσ_(imp)(T,Δ) for the various options define a set of values which can beplotted against the corresponding T and delta axes. A sample plot isillustrated in FIG. 2.

According to one aspect of the invention, a parameterized volatilitysurface providing a measure of the implied volatility σ_(i) for a givendelta and T at a time index i, is defined as a function F of one or moresurface parameters β_(o.i) . . . β_(n.i), delta, and T:σ_(i)(Δ,T)=F(β_(0,i), . . . β_(n,i,) Δ,T)+e _(i)(Δ,T)  (Equ. 2)As will be appreciated, various scaling functions can be applied to thevalue of σ_(i). The error or noise term e_(i) is not technically acomponent of the volatility surface model itself but is shown herein toindicate that the modeled surface may only be an approximation of thevolatility values.

Prior to simulation, values for the parameters β_(o) . . . β_(n) aredetermined to define a volatility surface via the volatility surfacemodel which approximates the historical volatility data from a giventime index. Suitable values can be determined using an appropriateregression analysis. The residual factor e_(i)(Δ,T) can be defined forat least some of the option points as used to determine the surfaceparameter values as an offset of the source volatility point from thecorresponding point on the modeled volatility surface. FIG. 3 shows animplied volatility surface determined in accordance with Equation 5(discussed below) from a regression of the set of volatility data pointsof FIG. 2. The residual offset values can be subsequently used tocalibrate or adjust volatility values which are extracted from themodeled volatility surface.

The form of the surface parameterization function and the number ofdifferent β parameters can vary depending on implementation specifics.Greater numbers of surface parameters can provide a surface that moreclosely fits the sample points but will also increase the complexity ofthe model. Preferably, the implied volatility surface is defined withreference to the log of the implied volatility values and is a linear orpiecewise linear function having at least one constant or planer term,one or more linear or piecewise linear parameter functions of delta, andone or more linear or piecewise linear parameter functions of T

A most preferred form of the surface parameterization function, in whichthe volatility value is scaled according to a log function, is:ln σ_(i)(Δ,T)=β_(0,i)+β_(1.i)(Δ−x ₁)+β_(2.i)(T−x ₂)⁺+β_(3.i)(T−x ₃)⁺ +e_(i)(Δ,T)  (Equ. 3)where (x)⁺ is a piecewise linear function equal to x where x>0 andotherwise equal to zero, e_(i)(Δ,T) is a residual noise factor, and x₁,x₂ and x₃ are constant terms having values selected as appropriate toprovide an acceptable surface fit to the historical data in accordancewith user preferences and other criteria.

Suitable values for x₁, x₂ and x₃ can be determined experimentally byapplying the simulation technique disclosed herein using differentvalues of x₁ . . . x₃ and then selecting values which provide the mostaccurate result. A similar technique can be used to select appropriatesurface parameterizing functions for the simulation of other riskfactors characterized by multiple variables. In a specificimplementation, the following values have been found to provide verysuitable results:ln σ_(i)(Δ,T)=β_(0.i)+β_(1.i)(Δ−0.5)+β_(2.i)(T−4)⁺+β_(3.i)(T−24)⁺ +e_(i)(Δ,T)  (Equ. 4)with the values of T specified in months. Variations in the specificvalues used and the form of the equation can be made in accordance withthe type of security and risk factor at issue as well as various otherconsiderations which will be recognized by those of skill in the art.

Depending upon the type of derivative value at issue and the dataavailable, conversions or translations of derivative characteristicsmight be required prior to using that data in the surface-definingregression. In addition, some decisions may need to be made regardingwhich data values to use during the regression. Preferably, a set ofpredefined guidelines is used to determine how the values of the impliedvolatilities which are regressed to derive the surface parameters areselected and also to identify outlying or incomplete data points whichshould be excluded from the regression.

According to a particular set of guidelines, for each underlier, theimplied volatilities used in the analysis can be selected usingfollowing rules:

-   -   For each exchange traded European option on the underlier,        closing bid and ask implied volatilities along with        corresponding delta and term are identified    -   Deltas of implied volatilities for puts are converted to the        deltas of calls using put-call parity    -   Implied volatilities with missing bid or ask or volatilties with        delta <0.15 or delta >0.85 are excluded    -   Average of bid-ask spread is used as data point    -   For underliers without exchange tradable options, implied        volatilities of OTC options marked by traders are used        As those of skill in the art will recognize, other sets of        guidelines can alternatively be used depending upon the        circumstances, the instruments at issue, and the variables        against which the volatility values are plotted to define the        surface.

After the initial surface parameters β for the surface volatility modelare determined, the model can be used to simulate changes in optionprice volatility by evolving the values of the beta surface parametersduring simulation and applying the simulated β values to the surfaceparameterization function to define a corresponding simulated volatilitysurface. The implied volatility of an option during simulation can bedetermined by referencing the simulated volatility surface in accordancewith the values of T and delta for that option at that point in thesimulation.

Although a typical regression analysis can produce a surface whichmatches the source data points fairly well, as seen in FIG. 3, many ofthe actual implied volatilities which are used to determine the surfaceparameters do not fall on the parameterized surface, but instead areoffset from it by a certain residual amount. Accordingly, after thevolatility surface is beta-parameterized and simulated, it isrecalibrated back to the actual implied volatilities by determining theresidual offset e_(i)(Δ,T) from the parameterized surface for at leastsome of the source volatility points.

To extract the implied volatility for an individual option duringsimulation, the simulated price of the underlying security and the timebefore the option expires are used to determine a point on the simulatedvolatility surface (generated using the simulated surface parametervalues). The residual offset for that point is then calculated withreference to the calibration data, for example, by interpolating fromthe nearest neighbor calibration points. The value of the volatilitysurface point adjusted by the interpolated residual offset can then beapplied to the simulation option pricing model. Although the changes inthe calibration residuals could be analyzed and adjusted during thesimulation process, preferably the calibration residuals are assumed tobe constant in time for all generated scenarios.

Various techniques can be used to calculate the evolving values of the βparameters during simulation. Generally, the beta evolution function isa function g of one or more parameters a₁ . . . a_(j), a prior value ofbeta, and a corresponding noise component ε:β_(m,i) =g(a ₁ , . . . a _(j),β_(m,i-1),ε_(m,i)  (Equ. 5)Preferably, the beta evolution function g is a linear mean-reversionprocess that provides a simulated time series of each individual betaparameter. A preferred form of the reversion providing a change in thebeta value is:Δβ_(m.i) =a _(m)(θ_(m)−β_(m.i-1))+ν_(m)ε_(m.i)  (Equ. 6)where α is a mean-reversion speed, θ is a mean for the β_(m), ν is avalue for the volatility of β_(m), and ε is a random, pseudo-random, orother noise term.

The values of α, θ, and ν can be determined empirically, estimated, orthrough other means. A preferred method is to determine these valuesbased upon historical analysis. In particular, historical data forvarious prior days i (or other time increment) is analyzed to generate acorresponding historical volatility surface having respective surfaceparameter values β_(m.i). This analysis produces a time series of valuesfor each surface parameter β_(m). The time-varying sequence of β_(m) isthen analyzed to determine the corresponding historic mean θ_(m),mean-reversion speed α_(m), and mean reversion volatility ν_(m). Thesevalues can then be used in Equ. 6 to simulate future values of therespective β_(m).

In some instances, there may be an insufficient number of impliedvolatility points to fully regress the set and determine appropriatevalues for each surface parameter. Various conditions specifying aminimum number of points and compensation techniques for situations withfewer points can be used. These conditions are dependant upon thecharacteristics of the surface parameterizing function and the number ofbeta parameters at issues.

According to a particular set of conditions which can be used inconjunction with a surface parameterization of the form shown in Equ. 3,above, at least 8 implied volatility points should be present to run aregression to determine the four beta parameters. These 8 volatilitiesshould have at least 2 different deltas and one term longer than 10months. In cases when these requirements are not met, the surfaceparameterization function can be simplified for the regression to reducethe number of betas. For example, when there is only one impliedvolatility point, only β_(o), will be calculated and the values for theremaining betas can be set to the previous day's values. Otherconditions can be specified for use when determining the parameters ofthe beta evolution function. For example, in a historical analysis usingthe mean reversion formula of Equ. 6, the mean reversion speed α_(m) canbe set to 2 years if the calculated speed is negative.

The method for simulating a risk factor surface according to theinvention is summarized in the flowchart of FIG. 4. Initially aparametric model is selected which defines a risk factor surfaceaccording to a plurality of parameters β_(o) . . . β_(n) are (step 40).The values of the risk factor on a given clay for a set of instrumentsderivative from a given security are regressed against the risk factorsurface model to determine the starting values of the surface parametersβ_(o) . . . β_(n). (Step 41) A calibration residual is determined for atleast some of the points used to define the starting surface parameterswhich represents the difference between the source point value and thevalue indicated by the modeled surface. (Step 42).

Next the evolution of each of the parameters β_(o) . . . β_(n) issimulated using a beta-evolution function. The function is preferably alinear mean-reversion process based upon historically determined values,such as a historical average for beta, beta volatility, and meanreversion speed. (Step 43). The sequences of simulated β_(o) . . . β_(n)values define a simulated risk factor surface for each time index ofeach simulation run. The appropriate reference points from thesimulation, such as the value of an underlying security and the deltafor an option and the beta values are applied to the surfaceparameterization model to determine a corresponding risk factor value.(Step 44). A residual offset is determined for that point by applyingthe calibration data, for example via extrapolating from the calibrationresidual values of the nearest “real” points used during the calibrationprocess (step 45) and this offset is applied to the risk factor value tocalibrate it. (Step 46). The calibrated risk factor value is then usedin the derivative pricing model, along with other data, to determine asimulated value of the derivative instrument. (Step 47).

Simulation of the surface parameter values and various risk factors canbe done on-the-fly during simulation. Preferably, however, thesimulation is performed in two primary steps—risk-factor pre-simulationand model application. This embodiment is illustrated in FIG. 5.

Initially, all of the simulated beta factor values for each simulation“tick” of each simulation scenario are generated and stored inrespective parameter value matrices. The simulated evolving values ofother risk factors used in the option pricing model are also“pre-simulated” and stored in a corresponding risk-factor matrices. Suchrisk factors can include, for example, simulated interest and loan ratevalues. In addition, because the option price is dependent upon theprice of an underlying equity, the price of the underlying equity isalso simulated using an appropriate equity model to provide a simulatedequity price matrix.

After the surface parameters, risk factors, and equity prices, as wellas other values which may be necessary are precalculated, theprecalculated values are extracted synchronously across the variousmatrices and used to simulate the option price. In particular, for agiven time index of a specific simulation run, the corresponding betasurface parameters are obtained from the surface parameter matrices.These values, when applied to the volatility surface model, define thesimulated volatility surface.

The simulated equity price and relevant option parameters such as Δ andT are determined for the option being simulated, for example, withreference to the simulated equity price, prior simulated values for theoption, and possibly other data. The Δ and T values (or other suitablevalues depending on the manner in which the volatility surface defined)are applied to the simulated volatility surface and the volatility valueis obtained. This value is then adjusted in accordance with thevolatility surface calibration data to provide a value for the simulatedoption volatility at that particular point of the simulation.

Finally, the simulated option volatility along with the appropriate riskfactor values (extracted from the corresponding simulated risk factormatrices) are applied to the option pricing model to produce a simulatedoption price for the particular option at issue. This process isrepeated for each step of each simulation run and the results are storedin a simulated option price matrix. When multiple options are to besimulated, the process is repeated for each option to generatecorresponding simulated option pricing matrices.

A further aspect of the invention is directed to the manner in which theevolving beta values are determined. When a parametric mean-reversion orother beta-evolution function is used to simulate changes in the surfaceparameter values over time, appropriate values of the correspondingnoise term ε_(m) must be selected. Preferably, the values of ε_(m) areselected from a predefined set of “historical” residual values. This setcan be derived by solving the beta evolution function for a sequence ofbeta values generated from historic volatility data to determine thesequence of noise values which recreates the “historical” beta sequence.This historical bootstrapping technique is addressed in detail in U.S.patent application Ser. No. 09/896,660, filed Jun. 29, 2001 and entitled“Method And System For Simulating Risk Factors In Parametric ModelsUsing Risk Neutral Historical Bootstrapping.” The historicalbootstrapping technique disclosed in this application can be applied tovolatility surface modeling by treating the beta values as risk factorsand the beta evolution equation as the corresponding parametricsimulation model. The entire contents of this application is herebyexpressly incorporated by reference.

For the beta evolution function of Equ. 6, the historical sequences ofas well as the derived values of the mean, mean reversion speed, andbeta volatility are applied to the mean-reversion beta evolutionfunction to produce a sequence of historical residual values accordingto:

$\begin{matrix}{ɛ_{m,i} = {\frac{1}{\upsilon}\left( {\beta_{m,i} - {a\left( {\theta - \mspace{2mu}\beta_{m,{i - 1}}} \right)}} \right)}} & \left( {{Equ}.\mspace{14mu} 7} \right)\end{matrix}$

The values of the determined historical residuals ε_(m,i) can then usedin the parametric beta evolution model during simulation in place ofrandom noise component. Prior to simulation, the range of values of thehistorical residuals should be standardized to the range suitable forthe corresponding random component in the model, typically such that theempirical average E[ε]=0 and the variance var[ε]=1. To preservecorrelations which may exist between different sets of residuals fromthe historical sample, a linear standardization process can be appliedto each residual value series to provide a corresponding standardizedseries:ε_(m,i) ′=k ₁ε_(m,i) +k ₂  (Equ. 8)where the values of k¹ and k² are selected to provide E[ε_(i)′]=0 andvar[ε_(i)′]=1 for the given series of ε_(m,i) at issue (and may bedifferent for different series). During simulation of the evolvingvalues of beta, values of ε_(m,i) are selected, preferably at random, tobe used in the beta-evolution function. To preserve cross-correlationsbetween the beta values, a single random index value is generated andused to select the historical residual value from the set of residualscorresponding to each beta parameter.

After the sets of historical residuals for the beta values aregenerated, the sets can be further processed by applying one or morebootstrapping techniques to account for certain deficiencies in thesource data, adjust the statistical distribution, increase the number ofavailable samples, or a combination of these or other factors prior tosimulation. To preserve correlations that may exist between thesequences of (standardized) historical residuals for each of the betaparameters, the same bootstrapping process should be applied to eachhistorical residual sequence.

For example, during a simulation of a large number of scenarios, thenumber of historical residuals used will typically greatly exceed theactual number of samples calculated from the historically derived betavalues. To increase the total number of historical residuals which areavailable, a multi-day bootstrap procedure can be used. A preferredbootstrapping technique is to sum a set of d randomly selected samplesand divide by the square-root of d to produce a new residual value:

$\begin{matrix}{ɛ^{''} = \frac{\sum\limits_{j = 1}^{d}\; ɛ_{j}^{\prime}}{\sqrt{d}}} & \left( {{Equ}.\mspace{14mu} 9} \right)\end{matrix}$This increases the total number of samples by a power of d (at the costof reducing kurtosis, the fourth moment of the statistical distribution,for higher values of d). Preferably, a two-day bootstrapping is used.For a 250 day history, this process produces a sequence of up to250*250=62,500 samples to draw on. Moreover, the low value of n=2 doesnot significantly reduce any fat-tail which may be present in thedistribution.

Other pre-simulation bootstrapping procedures can be performed, such assymmetrizing the distribution of residuals to permit both increasing anddecreasing beta value evolution if the source data provides betas whichshift primarily in only one direction. A symmetrized set can begenerated by randomly selecting two residual values i and j andcombining them as:

$\begin{matrix}{ɛ^{''} = \frac{ɛ_{i}^{\prime} - ɛ_{j}^{\prime}}{\sqrt{2}}} & \left( {{Equ}.\mspace{14mu} 10} \right)\end{matrix}$

Various other bootstrapping techniques known to those of skill in theart can also be used and more than one modification to the originallyderived set of historical residuals can be performed prior to thesimulation.

The methodology discussed above allows volatility a surface to bedefined for options on a security by determining a series of betasurface parameters associated with the historical performance of theoption and/or the underlying security. The methodology can also be usedto develop a volatility surface model for basket options, options onsector indexes, and other options which are based on multiple underlyingsecurities (all of which are generally referred to herein as “basketoptions” for simplicity).

In one embodiment, the surface parameters for the basket option aredetermined using historical data in a manner similar to that for optionsbased upon a single security. However, it can often be difficult toobtain a historical time series of implied volatilities based on OTCbaskets or sector indexes.

A further aspect of the invention provides a method for determining thesurface parameters of a volatility surface model for basket optionsdirectly from the surface parameters of the individual componentsecurities on which the basket is based. Similar to Equ. 2, above, thevolatility surface for basket options can be generally expressed as:σ_(B)(Δ,T)=F(β_(B,o), . . . β_(B,n) ,Δ,T)+e(Δ,T)  (Equ. 11)where σ_(B) is the volatility for basket B, β_(B,o), . . . β_(B,n) arethe parameters for the respective volatility surface model, and Δ, T ande are as defined above (but for the basket). According to this aspect ofthe invention, the values for β_(B,o), . . . β_(B,n) are deriveddirectly from the surface parameters for options on the N componentsecurities of the basket, e.g.:β_(B,x) =F _(k=1) ^(N)(β_(o,k), . . . ,β_(n,k) Δ,T, . . . )  (Equ. 12)Because the surface parameters for the components of a basket willtypically be calculated before the basket values are required, andadditional values which may be needed to relate the component parametersto the surface model parameters are also easy to determine,implementation of the present methodology in a simulation can be donewith minimal additional overhead. A specific most preferred relationshipbetween the basket option surface parameters and the surface parametersof the individual components is described below. However, otherrelationships can also be derived and this aspect of the inventionshould not be considered as being limited solely to the relationship(s)disclosed herein.

Initially, the price at a time t of a basket having fixed number ofshares for each component i can be defined as:

$\begin{matrix}{{B(t)} = {\sum\limits_{i = 1}^{n}\;{n_{i}{S_{i}(t)}{C_{i}(t)}}}} & \left( {{Equ}.\mspace{14mu} 13} \right)\end{matrix}$where B is the basket price, n_(i) is the number of shares of thecomponent i of the basket option, S_(i) is the price of component i in anative currency and C_(i) is an exchange rate between a currency forcomponent i and the currency in which the basket options are priced.

The price of the basket at a time t₂ relative to the price at a time t₁can then be written as:

$\begin{matrix}{{B\left( t_{2} \right)} = {{B\left( t_{1} \right)}{\sum\limits_{i}\;{{{\overset{\sim}{w}}_{i}\left( t_{i} \right)}\frac{{S_{i}\left( t_{2} \right)}{C_{i}\left( t_{2} \right)}}{{S_{i}\left( t_{1} \right)}{C_{i}\left( t_{i} \right)}}}}}} & \left( {{Equ}.\mspace{14mu} 14} \right)\end{matrix}$where {tilde over (w)}_(i)(t) is an effective spot rate for a componenti at a time t. Although various definitions for spot rate could be used,preferably, {tilde over (w)}_(i)(t) is defined as:

$\begin{matrix}{{\overset{\sim}{w}(t)} = \frac{n_{i}{S_{i}(t)}{C_{i}(t)}}{B(t)}} & \left( {{Equ}.\mspace{14mu} 15} \right)\end{matrix}$For values which change in accordance with a geometrical Brownian motionprocess, the following is a valid approximation:

$\begin{matrix}{{{\sum\limits_{i}\;{\lambda_{i}c_{i}}} = {{\mathbb{e}}^{i^{\sum\;{\lambda_{i}\log\mspace{14mu} c_{i}}}} + {O\left( {{\lambda_{i}\left( {c_{i} - 1} \right)}{\lambda_{i}\left( {c_{j} - 1} \right)}} \right)}}}{{{provided}\mspace{14mu}{that}\mspace{14mu}{\sum\limits_{i}\;\lambda_{i}}} = {1\mspace{14mu}{and}\mspace{14mu}{{1 - c_{i}}}{\operatorname{<<}1}}}} & \left( {{Equ}.\mspace{14mu} 16} \right)\end{matrix}$

For purposes of the present invention, changes in the volatility surfacefor basket options are considered to be subject to a geometricalBrownian motion process. Thus, using the approximation of Equation 16,and recognizing that

$\begin{matrix}{{\sum\limits_{i}\;{{\overset{\sim}{w}}_{i}(t)}} = {{1\mspace{14mu}{and}\mspace{14mu}\frac{{S_{i}\left( t_{2} \right)}{C_{i}\left( t_{2} \right)}}{{S_{i}\left( t_{1} \right)}{C_{i}\left( t_{1} \right)}}} \approx 1}} & \left( {{Equ}.\mspace{14mu} 17} \right)\end{matrix}$Equation 13 can be rewritten using a Taylor series expansion as thefollowing:

$\begin{matrix}{{{B\left( t_{2} \right)} \approx {{B\left( t_{1} \right)}{\mathbb{e}}^{\sum\limits_{i}\;{{{\overset{\sim}{w}}_{i}{(t_{1})}}{\log{(\frac{{S_{i}{(t_{2})}}{C_{i}{(t_{2})}}}{{S_{i}{(t_{1})}}{C_{i}{(t_{1})}}})}}}}}} = {{B\left( t_{1} \right)}{\prod\limits_{i}\;\left( \frac{{S_{i}\left( t_{2} \right)}{C_{i}\left( t_{2} \right)}}{{S_{i}\left( t_{1} \right)}{C_{i}\left( t_{1} \right)}} \right)^{w_{i}{(t_{i})}}}}} & \left( {{Equ}.\mspace{14mu} 16} \right)\end{matrix}$

A further simplifying assumption, suitable for many simulationscenarios, is that the implied volatility of a basket is dependent onlyon the implied volatility of basket components that have the same deltaand T. In these conditions, the basket volatility can be defined as:

$\begin{matrix}{{\sigma_{B}^{2}\left( {\Delta,T} \right)} = {\sum\limits_{i,j}\;{{\overset{\sim}{w}}_{i}{\overset{\sim}{w}}_{j}{\quad\left( {{p_{s_{i}s_{j}}{\sigma_{s_{i}}\left( {\Delta,T} \right)}{\sigma_{s_{i}}\left( {\Delta,T} \right)}} + {p_{s_{i}c_{j}}{\sigma_{s_{i}}\left( {\Delta,T} \right)}{\sigma_{c_{i}}\left( {\Delta,T} \right)}} + {p_{c_{i}s_{j}}{\sigma_{c_{i}}\left( {\Delta,T} \right)}{\sigma_{s_{j}}\left( {\Delta,T} \right)}} + {p_{c_{i}c_{j}}{\sigma_{c_{i}}\left( {\Delta,T} \right)}{\sigma_{c_{j}}\left( {\Delta,T} \right)}}} \right)}}}} & \left( {{Equ}.\mspace{14mu} 17} \right)\end{matrix}$

where σ_(Si)(Δ,T) is the implied volatilities of a components i quotedin a native currencies, σ_(Ci)(Δ,T) is an implied volatility of theexchange rates for the native currency component i, and ρ_(SiSj) are thecorresponding correlations between basket components i and j.

Substituting the value of the basket volatility into the parameterizedsurface model, such as in Equs. 2-4, allows the surface parameters forthe basket to be determined directly from the surface parameters of thebasket component. For example applying the volatility approximation ofEqu. 17 to the model of Equ. 4 and substituting σ_(Si)(Δ,T) with thesurface model and surface parameters for the component i provides:

$\begin{matrix}{{\mathbb{e}}^{{2\beta_{B,0}} + {2{\beta_{B,1}{({\Delta - 0.5})}}} + {2\beta_{B,2}} + {{({4 - T})}^{+}2{\beta_{B,3}{({24 - T})}}^{+}}} = {\underset{i,j}{{\quad\quad}\sum}\;{\overset{\sim}{w}}_{i}{\overset{\sim}{w}}_{j} \times \left\lbrack \begin{matrix}{{\rho_{s_{i}s_{j}}{\mathbb{e}}^{{({\beta_{oj} + \beta_{oj}})} + {({{{({\beta_{1j} + \beta_{1j}})}{({\Delta - 0.5})}} + {{({\beta_{2i} + \mspace{2mu}\beta_{2j}})}{({T - 4})}^{+}{({\beta_{3j} + \mspace{2mu}\beta_{3j}})}{({T - 24})}^{+}}}}}} +} \\{{\rho_{s_{i}c_{j}}{\mathbb{e}}^{\beta_{oj} + \mspace{2mu}{\beta_{1j}{({\Delta - 0.5})}} + \mspace{2mu}{\beta_{2j}{({T - 4})}}^{+} + {\beta_{i}^{3}{({T - 24})}}^{+}}{\sigma_{c_{i}}\left( {\Delta,T} \right)}} +} \\{{\rho_{c_{i}s_{j}}{\mathbb{e}}^{\beta_{oi} + {\beta_{1i}{({\Delta - 0.5})}} + {\beta_{2i}{({T - 4})}}^{+} + {\beta_{3i}{({T - 24})}}^{+}}{\sigma_{c_{i}}\left( {\Delta,T} \right)}} +} \\{\rho_{c_{i}c_{j}}{\sigma_{c_{i}}\left( {\Delta,T} \right)}{\sigma_{c_{j}}\left( {\Delta,T} \right)}}\end{matrix} \right\rbrack}} & \left( {{Equ}.\mspace{14mu} 18} \right)\end{matrix}$To determine the volatility model surface parameters for the basketdirectly from the volatility model surface parameters for the componentsof the basket, Equ. 18 can be solved for the surface parameter at issue.As will be appreciated, the mathematical solution can be somewhatcomplex. Reasonable estimates can be used to simplify a surfaceparameter relational equation, such as Equ. 18, in order to solve forthe basket surface parameters.

For example, β_(B,o) can be estimated substituting Δ=0.5 and T=24,eliminating the piecewise linear terms in the most preferred form of thesurface model, as expressed in Equ. 4 above. The result of such asubstitution yields:

$\begin{matrix}{\beta_{B,0} = {\frac{1}{2}\log{\quad\left( {\sum\limits_{i,j}\;{{\overset{\sim}{w}}_{i}{\overset{\sim}{w}}_{j}\left. \quad\left( \begin{matrix}{{\rho_{s_{i}s_{j}}{\mathbb{e}}^{\beta_{o,i} + \beta_{o,j}}} + {\rho_{S_{i}C_{j}}{\mathbb{e}}^{\beta_{o}j}{\sigma_{C_{j}}\left( {{.5},24} \right)}} +} \\{{\rho_{C_{i}S_{j}}{\mathbb{e}}^{\beta_{o,i}}{\sigma_{C_{i}}\left( {{.5},24} \right)}} + {\rho_{C_{i}C_{j}}{\sigma_{C_{j}}\left( {{.5},24} \right)}{\sigma_{C_{j}}\left( {{.5},24} \right)}}}\end{matrix} \right) \right)}} \right.}}} & \left( {{Equ}.\mspace{14mu} 19} \right)\end{matrix}$Similarly, estimates of β_(B,1 . . . 3) can be obtained from Equ. 18 bysubstituting (0, 24), (0.5, 23), and (0.5, 3), respectively, for (Δ,T).

The relationships between the surface parameters of the basketvolatility surface and the surfaces for the components can be simplifiedfurther for situations where all of the basket components arerepresented in the same currency, (i.e. σ_(Ci)≡0). Under this condition,specifying the values of the basket volatility model surface parameterscan be written in compact form as:

$\begin{matrix}{\mspace{79mu}{\beta_{B.o} = {\frac{1}{2}{\log\left( {\sum\limits_{i,j}\;{{\overset{\sim}{w}}_{i}{\overset{\sim}{w}}_{j}\rho_{ij}{\mathbb{e}}^{\beta_{o,i} + \beta_{o,j}}}} \right)}}}} & \left( {{Equ}.\mspace{14mu} 20} \right) \\{\mspace{79mu}{\beta_{B{.1}} = {\log\left( \frac{\sum\limits_{i,j}\;{{\overset{\sim}{w}}_{i}{\overset{\sim}{w}}_{j}\rho_{ij}{\mathbb{e}}^{\mspace{2mu}{\beta_{o,i} + \beta_{o,j}}}}}{\sum\limits_{i,j}\;{{\overset{\sim}{w}}_{i}{\overset{\sim}{w}}_{j}\rho_{ij}{\mathbb{e}}^{\beta_{o,i} + \beta_{o,j} + {{(\mspace{2mu}{\beta_{1,j} + \beta_{1,j}})}/2}}}} \right)}}} & \left( {{Equ}.\mspace{14mu} 21} \right) \\{\mspace{79mu}{\beta_{B{.3}} = {\frac{1}{2}{\log\left( \frac{\sum\limits_{i,j}\;{{\overset{\sim}{w}}_{i}{\overset{\sim}{w}}_{j}\rho_{ij}{\mathbb{e}}^{\mspace{2mu}{\beta_{o,i} + \beta_{{o.},} + \beta_{3,i} + \beta_{3,j}}}}}{\sum\limits_{i,j}\;{{\overset{\sim}{w}}_{i}{\overset{\sim}{w}}_{j}\rho_{ij}{\mathbb{e}}^{\mspace{2mu}{\beta_{o,i} + \beta_{o,j}}}}} \right)}}}} & \left( {{Equ}.\mspace{14mu} 22} \right) \\{\beta_{B{.2}} = {{{- 21}\mspace{2mu}\beta_{B,3}} + {\frac{1}{2}{\log\left( \frac{\sum\limits_{i,j}\;{{\overset{\sim}{w}}_{i}{\overset{\sim}{w}}_{j}\rho_{ij}{\mathbb{e}}^{\mspace{2mu}{\beta_{o,i} + \beta_{o,j} + \beta_{2,i} + \beta_{2,j} + {21{({\beta_{3,i} + \beta_{3,j}})}}}}}}{\sum\limits_{i,j}\;{{\overset{\sim}{w}}_{i}{\overset{\sim}{w}}_{j}\rho_{ij}{\mathbb{e}}^{\mspace{2mu}{\beta_{o,i} + \beta_{o,j}}}}} \right)}}}} & \left( {{Equ}.\mspace{14mu} 23} \right)\end{matrix}$

It should be appreciated that the above discussion presents a mostpreferred form for determining the surface parameter values for use inmodeling the volatility surface for basket options from the parametervalues of the basket components. This form results from variousassumptions which may not be appropriate under all circumstances.However, the general methodology as presented herein for generating therelational equations between the basket surface parameter values and thesurface parameters of the components can be used under differentcircumstances and appropriate changes and derivation techniques will beapparent to those of skill in the art.

The present invention can be implemented using various techniques. Apreferred method of implementation uses a set of appropriate softwareroutines which are configured to perform the various method steps on ahigh-power computing platform. The input data, and the generatedintermediate values, simulated risk factors, priced instruments, andportfolio matrices can be stored in an appropriate data storage area,which can include both short-term memory and long-term storage, forsubsequent use. Appropriate programming techniques will be known tothose of skill in the art and the particular techniques used depend uponimplementation details, such as the specific computing and operatingsystem at issue and the anticipated volume of processing. In aparticular implementation, a Sun OS computing system is used. Thevarious steps of the simulation method are implemented as C++ classesand the intermediate data and various matrices are stored usingconventional file and database storage techniques.

While the invention has been particularly shown and described withreference to preferred embodiments thereof, it will be understood bythose skilled in the art that various changes in form and details can bemade without departing from the spirit and scope of the invention.

1. A computer processor-implemented method for pricing a sector indexoption, the method comprising: retrieving by a computer processor avolatility surface model from memory based at least in part onhistorical data for options of a given security, the volatility surfacemodel defining a volatility surface using a plurality of surfaceparameters β_(o) . . . β_(n), where n is an integer greater than orequal to zero, wherein the surface model has a form: σ(Δ,T)=F(β_(o), . .. ,β_(n),Δ,T) where: σ is a measure of the volatility for an option witha given Δ and T, F is a function of Δ, T and the surface parametersβ_(o) . . . β_(n), Δ is a ratio of change in option price to change inunderlying instrument price, and T is an option term remaining;providing by the computer processor values for surface parametersβ_(o,k) . . . β_(n,k)1<k≦N that define, for each particular componentinstrument k of the sector index option, a respective volatility surfacevia the surface model for options on that component instrument;determining by the computer processor values for surface parametersβ_(B,o) . . . β_(B,n) defining a volatility surface for the sector indexoption using the surface parameters β_(o,k) . . . β_(n,k) associatedwith each component instrument; extracting by the computer processor avolatility value from the volatility surface for the sector index optiondefined by surface parameters β_(B,o) . . . β_(B,n); and determining bythe computer processor a price of the sector index option based on theextracted volatility value and at least one option pricing model storedin memory.
 2. The method of claim 1 wherein the surface model is of theformln σ(Δ,T)=β_(o)+β₁(Δ−x ₁)+β₂(T−x ₂)⁺+β₃(T−x ₃)⁺ where x₁, x₂ and x₃ areconstant terms and where (x)⁺ is a piecewise linear function equal to xif x>o and equal to o otherwise.
 3. An apparatus for pricing a sectorindex option, comprising: a memory; a computer processor disposed incommunication with said memory, and configured to issue a plurality ofprocessing instructions stored in the memory, wherein the instructionsare executable by the computer processor to: retrieve a volatilitysurface model from memory based at least in part on historical data foroptions of a given security, the volatility surface model defining avolatility surface using a plurality of surface parameters β_(o) . . .β_(n), where n is an integer greater than or equal to zero and where thesurface model has a form:σ(Δ,T)=F(β_(o), . . . ,β_(n) ,Δ,T)  where: σ is a measure of thevolatility for an option with a given Δ and T, F is a function of Δ, Tand the surface parameters β_(o) . . . β_(n), Δ is a ratio of change inoption price to change in underlying instrument price, and T is anoption term remaining; provide values for surface parameters β_(o,k) . .. β_(n,k)1<k≦N that define, for each particular component instrument kof the sector index option, a respective volatility surface via thesurface model for options on that component instrument; determine valuesfor surface parameters β_(B,o) . . . β_(B,n) defining a volatilitysurface for the sector index option using the surface parameters β_(o,k). . . β_(n,k) associated with each component instrument; extract avolatility value from the volatility surface for the sector index optiondefined by surface parameters β_(B,o) . . . β_(B,n); and determine aprice of the sector index option based on the extracted volatility valueand at least one option pricing model stored in memory.
 4. The apparatusof claim 3, wherein the surface model is of the form:ln σ(Δ,T)=β_(o)+β₁(Δ−x ₁)+α₂(T−x ₂)⁺+β₃(T−x ₃)⁺ where x₁, x₂ and x₃ areconstant terms and where (x)⁺ is a piecewise linear function equal to xif x>o and equal to o otherwise.
 5. The apparatus of claim 4, whereinx₁, x₂ and x₃ are substantially equal to 0.5, 4.0, and 24 respectively.6. The apparatus of claim 4, wherein${\beta_{B,0} = {\frac{1}{2}{\log\left( {\sum\limits_{i,j}\;{\overset{\sim}{w_{i}}{\overset{\sim}{w}}_{j}\rho_{ij}{\mathbb{e}}^{\beta_{0,i} + \beta_{0,j}}}} \right)}}};$${\beta_{B,1} = {\log\left( \frac{\sum\limits_{i,j}\;{{\overset{\sim}{w}}_{i}{\overset{\sim}{w}}_{j}\rho_{ij}{\mathbb{e}}^{\beta_{0,i} + \beta_{0,j}}}}{\sum\limits_{i,j}\;{{\overset{\sim}{w}}_{i}{\overset{\sim}{w}}_{j\;}\rho_{ij}{\mathbb{e}}^{\beta_{0,i} + \beta_{0,j} + {{({\beta_{1,i} + \beta_{1,j}})}/^{\;}}}2}} \right)}};$${\beta_{B,2} = {\frac{1}{2}{\log\left( \frac{\sum\limits_{i,j}\;{\overset{\sim}{w_{i}}{\overset{\sim}{w}}_{j}\rho_{ij}{\mathbb{e}}^{\beta_{0,i} + \beta_{0,j} + \beta_{3,i} + \beta_{3,j}}}}{\sum\limits_{i,j}\;{{\overset{\sim}{w}}_{i}{\overset{\sim}{w}}_{j}\rho_{ij}{\mathbb{e}}^{\beta_{0,i} + \beta_{0,j}}}} \right)}}};\;{and}$${\beta_{B,3} = {{{- 21}\beta_{B,3}} + {\frac{1}{2}{\log\left( \frac{\sum\limits_{i,j}\;{{\overset{\sim}{w}}_{i}{\overset{\sim}{w}}_{j}\rho_{ij}{\mathbb{e}}^{\beta_{0,i} + \beta_{0,j} + \beta_{3,i} + \beta_{3,j}}}}{\sum\limits_{i,j}\;{{\overset{\sim}{w}}_{i}{\overset{\sim}{w}}_{j}\rho_{ij}{\mathbb{e}}^{\beta_{0,i} + \beta_{0,j}}}} \right)}}}};$where {tilde over (ω)}_(i)(t) is an effective spot rate for a componenti at a time t, and ρ_(ij) is a correlation between a price of instrumenti and a price of instrument j.
 7. The apparatus of claim 6, wherein:${{{\overset{\sim}{w}}_{i}(t)} = \frac{n_{i}{S_{i}(t)}{C_{i}(t)}}{B(t)}},$where n_(i) is a number of shares of the component i of the sector indexoption, S_(i) is a price of component i in a native currency, C_(i) isan exchange rate between the native currency for component i and acurrency in which the sector index options are priced, and B(t) is asector index price at time t.
 8. The apparatus of claim 7, wherein arelationship between the surface parameters β_(B,o) . . . β_(B,n)defining the volatility surface of the sector index option and thesurface parameters β_(o,k) . . . β_(n,k) defining volatility surfacesfor options on the components of the sector index option, can beexpressed as:${\mathbb{e}}^{{2\beta_{B,0}} + {2{\beta_{B,1}{({\Delta - 0.5})}}} + {2{\beta_{B,2}{({T - 4})}}^{+}} + {2{\beta_{B,3}{({T - 24})}}^{+}}} = {\sum\limits_{i,j}\;{{\overset{\sim}{w}}_{i}{\overset{\sim}{w}}_{j} \times \begin{bmatrix}{{\rho_{s_{i},s_{j}}{\mathbb{e}}^{{({\beta_{0,i} + \beta_{0,j}})} + {{({\beta_{1,i} + \beta_{1,j}})}{({\Delta - 0.5})}} + {{({\beta_{2,i} + \beta_{2,j}})}{({T - 4})}^{+}} + {{({\beta_{3,i} + \beta_{3,j}})}{({T - 24})}^{+}}}} +} \\{\rho_{s_{i}},{{c_{j}{\mathbb{e}}^{\beta_{0,i} + {\beta_{1,i}{({\Delta - 0.5})}} + {\beta_{2,i}{({T - 4})}}^{+} + {\beta_{3,i}{({T - 24})}}^{+}}{{\sigma_{c}}_{j}\left( {\Delta,T} \right)}} +}} \\{\rho_{c_{i}},_{s_{j}}{{{\mathbb{e}}^{\beta_{0,j} + {\beta_{1,j}{({\Delta - 0.5})}} + {\beta_{2,j}{({T - 4})}}^{+} + {\beta_{3,j}{({T - 24})}}^{+}}{\sigma_{c_{i}}\left( {\Delta,T} \right)}} +}} \\{\rho_{c_{i},c_{j}}{\sigma_{c_{i}}\left( {\Delta,T} \right)}{\sigma_{c_{j}}\left( {\Delta,T} \right)}}\end{bmatrix}}}$ where σ_(C) _(i) (Δ,T) is an implied volatility of theexchange rates between the native currency for component i and thesector index pricing currency and ρ_(Y,Z) is a correlation between Y andZ.
 9. A sector index pricing processor-readable non-transitory mediumstoring a plurality of processing instructions for pricing a sectorindex option comprising processor-executable instructions by a processorto: retrieve a volatility surface model based at least in part onhistorical data for options of a given security, the volatility surfacemodel defining a volatility surface using a plurality of surfaceparameters β_(o) . . . β_(n), where n is an integer greater than orequal to zero, and where the surface model has a formσ(Δ,T)=F(β_(o), . . . ,β_(n) Δ,T)  where: σ is a measure of thevolatility for an option with a given Δ and T, F is a function of Δ, Tand the surface parameters β_(o) . . . β_(n), Δ is a ratio of change inoption price to change in underlying instrument price, and T is anoption term remaining; provide values for surface parameters β_(o,k) . .. β_(n,k)1<k≦N that define, for each particular component instrument kof the sector index option, a respective volatility surface via thesurface model for options on that component instrument; determine valuesfor surface parameters β_(B,o) . . . β_(B,n) defining a volatilitysurface for the sector index option using the surface parameters β_(o,k). . . β_(n,k) associated with each component instrument; extract avolatility value from the volatility surface for the sector index optiondefined by surface parameters β_(B,o) . . . β_(B,n); and determine aprice of the sector index option based on the extracted volatility valueand at least one option pricing model stored in memory.
 10. The mediumof claim 9, wherein the surface model is of the form:ln σ(Δ,T)=β_(o)+β₁(Δ−x ₁)+β₂(T−x ₂)⁺+β₃(T−x ₃)⁺ where x₁, x₂ and x₃ areconstant terms and where (x)⁺ is a piecewise linear function equal to xif x>o and equal to o otherwise.
 11. The medium of claim 10, wherein x₁,x₂ and x₃ are substantially equal to 0.5, 4.0, and 24 respectively. 12.The medium of claim 11, wherein${\beta_{B,0} = {\frac{1}{2}{\log\left( {\sum\limits_{i,j}\;{\overset{\sim}{w_{i}}{\overset{\sim}{w}}_{j}\rho_{ij}{\mathbb{e}}^{\beta_{0,i} + \beta_{0,j}}}} \right)}}};$${\beta_{B,1} = {\log\left( \frac{\sum\limits_{i,j}\;{{\overset{\sim}{w}}_{i}{\overset{\sim}{w}}_{j}\rho_{ij}{\mathbb{e}}^{\beta_{0,i} + \beta_{0,j}}}}{\sum\limits_{i,j}\;{{\overset{\sim}{w}}_{i}{\overset{\sim}{w}}_{j\;}\rho_{ij}{\mathbb{e}}^{\beta_{0,i} + \beta_{0,j} + {{({\beta_{1,i} + \beta_{1,j}})}/^{\;}}}2}} \right)}};$${\beta_{B,2} = {\frac{1}{2}{\log\left( \frac{\sum\limits_{i,j}\;{\overset{\sim}{w_{i}}{\overset{\sim}{w}}_{j}\rho_{ij}{\mathbb{e}}^{\beta_{0,i} + \beta_{0,j} + \beta_{3,i} + \beta_{3,j}}}}{\sum\limits_{i,j}\;{{\overset{\sim}{w}}_{i}{\overset{\sim}{w}}_{j}\rho_{ij}{\mathbb{e}}^{\beta_{0,i} + \beta_{0,j}}}} \right)}}};\;{and}$${\beta_{B,3} = {{{- 21}\beta_{B,3}} + {\frac{1}{2}{\log\left( \frac{\sum\limits_{i,j}\;{{\overset{\sim}{w}}_{i}{\overset{\sim}{w}}_{j}\rho_{ij}{\mathbb{e}}^{\beta_{0,i} + \beta_{0,j} + \beta_{3,i} + \beta_{3,j}}}}{\sum\limits_{i,j}\;{{\overset{\sim}{w}}_{i}{\overset{\sim}{w}}_{j}\rho_{ij}{\mathbb{e}}^{\beta_{0,i} + \beta_{0,j}}}} \right)}}}};$where {tilde over (ω)}_(i)(t) is an effective spot rate for a componenti at a time t, and ρ_(ij) is a correlation between a price of instrumenti and a price of instrument j.
 13. The medium of claim 12, wherein:${{{\overset{\sim}{w}}_{i}(t)} = \frac{n_{i}{S_{i}(t)}{C_{i}(t)}}{B(t)}},$where n_(i) is a number of shares of the component i of the sector indexoption, S_(i) is a price of component i in a native currency, C_(i) isan exchange rate between the native currency for component i and acurrency in which the sector index options are priced, and B(t) is asector index price at time t.
 14. The medium of claim 13, wherein arelationship between the surface parameters β_(B,o) . . . β_(n,k)defining the volatility surface of the sector index option and thesurface parameters β_(o,k) . . . β_(n,k) defining volatility surfacesfor options on the components of the sector index option, can beexpressed as:${\mathbb{e}}^{{2\beta_{B,0}} + {2{\beta_{B,1}{({\Delta - 0.5})}}} + {2{\beta_{B,2}{({T - 4})}}^{+}} + {2{\beta_{B,3}{({T - 24})}}^{+}}} = {\sum\limits_{i,j}\;{{\overset{\sim}{w}}_{i}{\overset{\sim}{w}}_{j} \times \begin{bmatrix}{{\rho_{s_{i},s_{j}}{\mathbb{e}}^{{({\beta_{0,i} + \beta_{0,j}})} + {{({\beta_{1,i} + \beta_{1,j}})}{({\Delta - 0.5})}} + {{({\beta_{2,i} + \beta_{2,j}})}{({T - 4})}^{+}} + {{({\beta_{3,i} + \beta_{3,j}})}{({T - 24})}^{+}}}} +} \\{\rho_{s_{i}},{{c_{j}{\mathbb{e}}^{\beta_{0,i} + {\beta_{1,i}{({\Delta - 0.5})}} + {\beta_{2,i}{({T - 4})}}^{+} + {\beta_{3,i}{({T - 24})}}^{+}}{{\sigma_{c}}_{j}\left( {\Delta,T} \right)}} +}} \\{\rho_{c_{i}},_{s_{j}}{{{\mathbb{e}}^{\beta_{0,j} + {\beta_{1,j}{({\Delta - 0.5})}} + {\beta_{2,j}{({T - 4})}}^{+} + {\beta_{3,j}{({T - 24})}}^{+}}{\sigma_{c_{i}}\left( {\Delta,T} \right)}} +}} \\{\rho_{c_{i},c_{j}}{\sigma_{c_{i}}\left( {\Delta,T} \right)}{\sigma_{c_{j}}\left( {\Delta,T} \right)}}\end{bmatrix}}}$ where σ_(C) _(i) (Δ,T) is an implied volatility of theexchange rates between the native currency for component i and thesector index pricing currency and ρ_(Y,Z) is a correlation between Y andZ.